54 



PEOFESSOE A. E. FOESYTH ON THE INTEGEATION OF 



Application to V 2 v = 0. 



31. When the method is applied to the potential equation, which is 



a + b + c = 



with the present notation, the substitution of values (say of a and b) is required to 

 lead to a result evanescent so far as the determination of coefficients of the second 

 order is concerned. The substitution gives 



til 



so that we must have 



(In dm dn 



~ + ~ -- * ~ + c 



P~ + 2 2 + 1 = 0, 



m rf/i 



, -- (7i- 

 ay 1 ay 



rW <7?i . rfm 



-- PT~ 



tl>: L ax 



and the differentiations with regard to x and to y in these relations are effected on 

 the supposition that the unexpressed variable u is constant. 

 The subsidiary simultaneous system thus is 



o 



dl dn dm dn 



dx " dx dy * dy 



dl dn dm dn 



dv 



= t + np 



dv 



= m + nq 



When HAMBURGER'S method, as expounded in the second of his memoirs already 

 quoted ( 30), is applied to this system, it is found that the algebraical equations for 

 the determination of the subsidiary multipliers are inconsistent with one another 

 unless all the multipliers are zero ; there is then a null result. Accordingly integrable 

 combinations must be obtained otherwise. 



Now the general solution of the equation 



is given by 



P 3 + q" + 1 = 



p = constant, q = constant, 

 z px qy =. constant, 



